Consider the following set of data on the world record times R , in seconds, in the 100 meter dash for both men and women at various times t over the last century.
|
|
1940 |
1950 |
1960 |
1970 |
1980 |
1990 |
2000 |
|---|---|---|---|---|---|---|---|
|
Men |
10.2 |
10.2 |
10.0 |
9.95 |
9.95 |
9.9 |
9.84 |
|
Women |
11.6 |
11.5 |
11.3 |
11.0 |
10.88 |
10.49 |
10.49 |
Regression analysis gives us the two linear functions that best fit these world record times:
for men:
for women:
where t represents the number of years since 1900. The corresponding scatterplot with the two lines superimposed is shown in Figure 4.1. Both lines are slanted downward, as we would expect, but the line for the women has a considerably larger negative slope. In fact, when we compare the slopes, we see that the slope
Figure 4.1
To answer this question, we have to find the point where the extensions of the two lines intersect one another. We can estimate this point numerically by trial-and-error (substitute different values for t into both equations until we find the time t when the women's world record is less than the men's record). For instance, we can try
The third approach is to think of the two equations as a pair of simultaneous linear equations . To do so, it is probably a good idea to rewrite both equations by bringing the variable terms to the same side of each equation by adding the appropriate quantities to both sides of each equation. When we do this, we obtain
Before trying to solve this pair of equations, we first need to consider some fundamental ideas on solving 40 systems of linear equations. We will come back to this situation later in this section.
Figure 4.2
Use trial-and-error to estimate the year, correct to one decimal place, in which the women's world record time in the 100 meter dash will surpass the men's record, assuming that the trends in the world records continue.
A pair of linear equations such as
(1)
(2)
is called a system of simultaneous linear equations or simply a system of linear equations . Its solution is a pair of values, one for x and the other for y , that satisfy both equations simultaneously . The solution to the system of Equations (1) and (2) is and (you will see below how we get it). To verify that this is indeed the solution, we substitute and into each of the two original equations in turn. For Equation (1), we have
and for Equation (2),
so these two values satisfy both equations and hence is the solution.
For now, we consider only systems of two linear equations in two unknowns. Geometrically, each of the two linear equations represents a line and every pair of numbers x and y that satisfies either of the two equations is a point (x , y ) on that line. A single pair of values for x and y that satisfy both equations simultaneously must be the point of intersection of the two lines, as shown in Figure 4.3. To graph these lines using a calculator or a computer program, it is typically necessary to solve for y in terms of x . For Equation (1), we first subtract x from both sides to get
and then divide both sides by 5 to obtain
For Equation (2), we first add 3y to both sides to get
and then subtract 11 from both sides, so that
When we divide both sides by 3, we have
From Figure 4.3, we see that the two lines seem to intersect at the point
Figure 4.3
Use the graphical approach to estimate the year in which the women's world record time in the 100 meter dash will surpass the men's record assuming that the trends in the world records continue.
We can always use this graphical approach to solve systems of two equations in two unknowns, but the best we can get is a reasonably accurate estimate of the solution, even after zooming in on the intersection point repeatedly. Worse, if we have a system of more than two equations in two unknowns, say a system of four equations in four unknowns, then such a geometric approach is impossible. Alternatively, we can solve a system of linear equations algebraically.
We begin by reviewing briefly some ideas on solving a system of linear equations algebraically, as a reminder and a demonstration of pencil-and-paper techniques. Later in this section, we will look at the far more efficient and easier methods that are used in practice today.
Typically, the algebraic approach involves one of two methods, the method of substitution and the method of elimination .
1. The method of substitution: Solve for one variable (say x ) in terms of the other variable (in this case, y ) using either of the two equations. Next, substitute the expression for that variable (x ) into the other equation, leaving a single equation in the other variable (y ). (This is usually straightforward if the coefficient of one of the variables is 1 or ; otherwise it can get fairly messy.) Then solve for the first variable. Finally, substitute its value back into either of the previous equations to find the value of the other variable. We illustrate this process in the following example.
Example 1 Solve the system of Equations (1) and (2) using the method of substitution.
Solution Since the coefficient of x in Equation (1) is 1, we use Equation (1) to solve for x in terms of y as
. (3)
We substitute this expression into Equation (2) to get
We apply the distributive law to obtain
Note that we have eliminated the variable x and now have a single equation in y only. We collect like terms to get
so that, when we divide both sides by
We now substitute
This is the same solution
2. The method of elimination: Add or subtract an appropriate multiple of one of the equations to the other equation to eliminate one of the variables.
Example 2 Solve the same system of linear equations
using the method of elimination.
Solution Suppose we choose to eliminate the variable x from the two equations. To do this, we multiply Equation (1) by
while Equation (2) is
(We did this to produce coefficients of x that are numerically equal, but of opposite sign.) If we add Equation (2) and Equation (4), the x terms cancel, leaving
so that, when we divide both sides by ,
,
as before. To solve for the other variable x , we now substitute this value for y into either Equation (1) or (2), say Equation (1), and get
or
so that, when we add 5 to both sides, we get
,
the same solution once more. Note that we would have obtained the same value for x if we had substituted
Alternatively, instead of eliminating x from Equations (1) and (2), we could eliminate the variable y . To do this, we multiply Equation (1) by 3 and Equation (2) by 5 to get:
Notice that one equation has the term
and so
Example 3 Solve the system of linear equations
for the world record times in the 100 meter dash for men and for women using (a) the method of substitution; (b) the method of elimination.
Solution a. From the two equations, it is obviously simpler to solve for R from either equation and substitute into the other equation than to solve for t . Using the first equation, we have
so when we substitute this into the second equation, we get
so that
Combining like terms, we then have
so that
or almost 142 years after 1900, which would be in late 2042. This value agrees with our graphical solution at the beginning of the section.
b. To solve the system of equations using the elimination method, we subtract the first equation from the second equation (to have a positive coefficient for the t -term) and so obtain
which becomes
and therefore
which is the identical solution.
If we have a system of three linear equations in three unknowns (say x , y , and z or
Most graphing calculators have the capability of solving systems of up to 99 equations in 99 unknowns at the push of a button. On some calculators, there is a SIMULT key for simultaneous equations; you typically enter the number of linear equations, and then enter the coefficients and the constant terms, and finally press Solve to get the solutions. On other calculators, you can solve a system of linear equations using the Solve command; you enter the list of equations, the list of variables, and press enter. On all scientific or more sophisticated calculators, you can solve systems of equations using matrix methods, as discussed below. The routines used by the calculators and by software packages to solve systems of linear equations all use matrix methods.
A matrix is any rectangular array of numbers, such as
The size, or dimension , of a matrix is measured by the number of rows (horizontally across) and the number of columns (vertically down) in the array. Since the matrix above has 3 rows and 4 columns, its dimension is 3 by 4, which we can write as
• square matrices , say 2 by 2 or 3 by 3 or 4 by 4, and
• column matrices that have only a single column, say 2 rows and 1 column or 4 rows and 1 column. Such column matrices are called column vectors or simply vectors .
Most of the ideas and methods we discuss extend in a very natural way to square matrices of size n by n , for any positive integer n and vectors of size n by 1, and many of these ideas can also extend to more general rectangular matrices.
Let's look at the system of linear equations:
(We consider this same system in Appendix I, where we find the solution algebraically to be
This matrix has 3 rows and 3 columns, so its dimension is 3 by 3, or
Since this vector has 3 rows and 1 column, its dimension is
These are the three unknowns we must determine.
The system of three linear equations in three unknowns is then equivalent to the simple matrix equation
For now, we only consider the mechanics of solving this matrix equation; later, we will look at some of the mathematical details. The solution X to the matrix equation
The solution to the matrix equation
is the matrix
assuming
To solve this matrix equation on the calculator, and so solve the corresponding system of linear equations, you must "name" each of the matrices in turn by giving its dimension and then entering the values for each position in each matrix. The dimension of matrix A is and the dimension of matrix B is . Finally, by selecting the appropriate names of the matrices, you have the calculator find
See the instruction manual for your calculator for more specific details on how to use its matrix features. We will discuss the meaning of matrix multiplication and the inverse matrix in more detail in Section 4.3; for now, we will focus on the details of solving systems of linear equations and some of the many applications that lead to such systems.
Incidentally, if any of the terms in one or more of the equations is missing, as in
Use the matrix features of your calculator to verify that
E xample 4 Use matrices to solve the system of linear equations
Solution The coefficient matrix and the matrix of constants are
and .
When we enter these matrices into the calculator and form the expression (or simply ), we find that the corresponding matrix of variables is
Correct to three decimal places, the solution to the system of equations is , , and
E xample 5 Use matrix methods to solve the system of linear equations
for the world record times in the 100 meter dash for men and for women.
Solution The coefficient matrix and the matrix of constants are
We enter these matrices into the calculator and form the expression to find that the corresponding matrix of variables is
Thus, when , the world record time for both men and women would be about
The value of using matrix methods is that they are just as easy to apply to much larger systems of linear equations, as we illustrate in the following example.
E xample 6 Use matrices to solve the following system of linear equations:
Solution For this system of four equations in four unknowns, the coefficient matrix and the matrix of constants are
When we enter these matrices into the calculator and form the expression , we find that the corresponding matrix of variables is
That is, correct to three decimal places, the solution to the system of equations is
1. Verify whether or not
2. Verify whether or not
3. Verify whether or not
4. Verify whether or not
5. What are the dimensions of each of the following matrices?
a.
b.
c.
Problems 6–8 refer to the matrix In each, two vectors are shown. One is the solution vector X and the other is the vector of constants B in the matrix equation
6. and
7. and
8. and
9. Determine, using your calculator or a software package, whether or not each of the following matrices have inverses.
a.
b.
c.
d.
e.
f.
Find the solution to each of the following systems of equations graphically.
10.
11.
12.
13.
14.
1 5 .
Find the solution, if it exists, to each of the following systems of equations using appropriate technology.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Systems of linear equations arise in many different ways. We begin with a rather routine application that is typical of the kind of problems you likely saw in high school algebra. However, our primary focus here will be on other kinds of applications and we will consider a much wider range of applications of these ideas later in this section.
E xample 1 One of the standard dinner offerings at Brookdale College's dining hall is meatloaf with mashed potatoes and green beans. The staff dietitian has to make up a meal with 475 calories, 36 grams of protein, and 55 grams of carbs. (These values are called the demand .) Each ounce of meatloaf has 75 calories, 7 grams of protein, and 6 grams of carbs. Each scoop of mashed potatoes has 60 calories, 2 grams of protein, and 10 grams of carbs. Each ounce of green beans has 10 calories, 1 gram of protein, and 2 grams of carbs. The dietician has to determine how many ounces of meatloaf, how many scoops of mashed potatoes, and how many ounces of green beans are needed to fulfil the demand.
a. Construct a system of three linear equations in three unknowns that represents a mathematical model for this situation.
b. Solve the system from part (a) using matrix methods to determine how much of each of the three items goes into a meal.
Solution a. Suppose that we let M represent the number of ounces of meatloaf needed P represent the number of scoops of mashed potatoes, and G represent the number of ounces of green beans. We can organize the given information in the following table
|
|
Meatloaf M |
Mashed Potatoes Ρ |
Green Beans G |
Demand |
|---|---|---|---|---|
|
Calories |
75 |
60 |
10 |
475 |
|
Protein |
7 |
2 |
1 |
36 |
|
Carbs |
6 |
10 |
2 |
55 |
This table sets the stage to write the needed system of equations and to get the entries in the matrices we will need to solve the system.
We look at each of the categories—calories, protein, and carbs—separately. The total number of calories in a meal is therefore made up of 75 calories from each of the M ounces of meatloaf, 60 calories from each of the P scoops of mashed potatoes, and 10 calories from each of the G ounces of green beans. That is,
Calories
Similarly, the total number of grams of protein is given by
Protein
and the total number of grams of carbs is given by
Carbs
Since the dietician needs to have a meal with 475 calories, 36 grams of protein, and 55 grams of carbs, these demand values give a system of three equations in three unknowns:
Calories:
Protein:
Carbs:
b. For this system, the coefficient matrix A is
and the matrix of constants (sometimes called the demand matrix ) is
The resulting matrix equation is
and the solution is
or, more meaningfully, the meal should consist of 4 ounces of meatloaf, 2 scoops of mashed potatoes, and 4 ounces of green beans.
How much effect does rounding the values have on the demand values?
Suppose that we have a set of data that falls into a roughly linear pattern. We can use linear regression to obtain the equation of the line that is the best fit to the data in the sense that it is the one line that comes closest to all the data points, as we discussed in Section 3.4. The process of applying the criterion that the desired line
where n is the number of data points and
We discuss summation notation using
Equations (1) and (2) are called the regression equations . We illustrate the process of creating these equations and solving them, using matrices, in the following example. Note that this is not intended as a substitute for using the linear regression features of your calculator or spreadsheet, but rather to demonstrate an important use of matrices and, simultaneously, to provide you with a deeper understanding of some of the issues we raised in the last section.
Example 2 Consider the set of data
|
X |
1 |
2 |
3 |
4 |
5 |
6 |
|---|---|---|---|---|---|---|
|
y |
10 |
22 |
31 |
43 |
54 |
62 |
a. Use regression techniques to find the equation of the regression line that fits the data.
b. Create the regression equations in which the coefficients a and b in
c. Solve the regression equations from part (b) using matrix methods and compare the results to those from part (a).
Solution a. When we enter these data values into a calculator or spreadsheet and select linear regression, we get
b. To calculate the necessary terms for the regression equations (1) and (2), we rewrite the table of data values in columns and add extra columns for the values of
|
x |
y |
|
|
|---|---|---|---|
|
1 |
10 |
1 |
10 |
|
2 |
22 |
4 |
44 |
|
3 |
31 |
9 |
93 |
|
4 |
43 |
16 |
172 |
|
5 |
54 |
25 |
270 |
|
6 |
62 |
36 |
372 |
|
|
|
|
|
In addition, we also record the sum of all the entries in each column in the bottom row. Thus, the sum of all the x values,
We now substitute these values for the sums into the regression equations (1) and (2). Equation (1) becomes
or, equivalently,
Similarly, Equation (2) becomes
or, equivalently,
c. To solve this system of linear equations using matrices, we introduce the matrix of coefficients and the matrix of constants
along with the matrix of unknowns,
Using a calculator, we find that the solution of the matrix equation is
That is,
In Section 3.4, we suggested that it often makes sense to scale down large data values to make the computations simpler. Otherwise, it is possible to get overflow errors when the size of the quantities involved exceed the capabilities of the technology being used. Suppose that the independent variable x that we used in Example 2 represents the number of years since 2000, so
Example 3 Consider the set of data
|
x |
2001 |
2002 |
2003 |
2004 |
2005 |
2006 |
|---|---|---|---|---|---|---|
|
y |
10 |
22 |
31 |
43 |
54 |
62 |
a. Find the equation of the regression line using a linear regression routine and compare the coefficients to those in Example 2.
b. Create the regression equations in which the coefficients a and b in
c. Solve the regression equations to obtain the equation of the regression line to fit this set of data.
d. Use both the regression line from Example 2 and the regression line from part (b) to predict the next term (when
Solution a. When we perform linear regression on this set of data, we get
b. As in Example 2, we rewrite the table of data values in columns and add extra columns for the values of
|
x |
y |
|
|
|---|---|---|---|
|
2001 |
10 |
4004001 |
20010 |
|
2002 |
22 |
4008004 |
44044 |
|
2003 |
31 |
4012009 |
62093 |
|
2004 |
43 |
4016016 |
86172 |
|
2005 |
54 |
4020025 |
108270 |
|
2006 |
62 |
4024036 |
124372 |
|
|
|
|
|
Notice how large the sums have become compared to the equivalent sums in Example 1. When we substitute these sums into the regression equations, Equation (1) becomes
or equivalently,
Similarly, Equation (2) becomes
In this case, the equations are considerably more complicated than those in Example 2 and, worse, the size of the coefficients could potentially lead to the overflow errors mentioned above.
c. To solve the regression equations
we introduce the matrix A of coefficients and the matrix B of constants
and .
Some calculators give the solution of the matrix equation
This gives the equation of the regression line as
d. To predict the "next" point
This is a reasonable prediction considering that the value given in the table in Example 2 when
which is fairly close to the value we got with the other equation. (The difference is likely due to rounding the coefficients.)
Clearly, scaling down the values as we did in Example 2 is a much easier, safer, and better approach.
The process of photosynthesis involves the chemical reaction in which carbon dioxide,
Perhaps the most common problem in elementary chemistry is that of balancing a chemical equation —that is, determining precisely how many molecules of each chemical are needed to make the equation "work". For photosynthesis, the balanced chemical equation turns out to be:
that is, it takes six molecules of carbon dioxide to combine with six molecules of water to produce one molecule of glucose and six molecules of oxygen. The reason that chemists describe this equation as being in balance is as follows. Consider the number of atoms of each element:
1. There are six atoms of carbon on the left (in
2. There are 12 atoms of hydrogen on the left (two atoms of hydrogen in each of the six molecules of
3. There are 18 atoms of oxygen on the left (two in each of the six molecules of carbon dioxide plus six more in the water) and there are 18 atoms of oxygen on the right (two in each of the six molecules of oxygen
The problem in chemistry is not looking at a balanced equation and seeing why it is in balance; rather, the problem is to determine the number of molecules of each substance that are needed to achieve balance. In chemistry, this is typically approached using a trial-and-error method, which is usually tedious and often confusing to many students. Alternatively, it turns out to be a simple application of our present methods—just set up a system of linear equations and solve it using matrix methods. We illustrate this systematic approach in several examples.
Example 4 Use matrix methods to balance the chemical equation for the photosynthesis reaction.
Solution In this chemical reaction, we can think of the glucose,
Equivalently, we can think of this reaction equation as
where we have isolated our target molecule
Carbon:
Hydrogen:
Oxygen:
This system of three linear equations in three unknowns is particularly simple and can actually be solved very easily using algebra. The first equation reduces to
or equivalently
If we add 2z to both sides of the equation and subtract 6 from both sides of the equation, we have
so that
Thus, our solution gives
as before.
Alternatively, we can solve this system of equations using matrix methods. We can write the system as the matrix equation
and .
The solution is
Therefore,
or, in more standard format,
as before.
Example 5 The main constituent of natural gas is methane. When natural gas burns in air, the methane combines with oxygen to produce carbon dioxide and water. That is, molecules of methane
a. the target is
b. the target is
Solution a. To produce one molecule of carbon dioxide
Equivalently, we can think of this reaction equation as
where we have isolated our target molecule
Carbon:
Hydrogen:
Oxygen:
This system of three linear equations in three unknowns can be written as the matrix equation
and .
The solution is
.
Since
or, in more standard format,
b. To produce one molecule of water,
or, equivalently,
where now the target molecule
Carbon:
Hydrogen:
Oxygen:
This new system of equations can also be written in matrix form as
Therefore, the balanced equation is
or equivalently
Because it makes little sense to have halves of molecules, we multiply both sides of this equation by 2 to eliminate the fractions and end up with the balanced equation
which is the same equation we had before.
In this example, we looked at one of the products of a reaction as our "target". Alternatively, we can think of one of the original reactants as our "target" in the process of balancing a chemical reaction. We illustrate this in the following example.
Example 6 When dimethyl ether,
Solution Suppose we have one molecule of dimethyl ether,
Equivalently, we can think of this reaction equation as
where we have isolated our target molecule
Carbon:
Hydrogen:
Oxygen:
We write this system of three linear equations in three unknowns as the matrix equation
The solution
and so the balanced equation is
We note that the system of equations that arose in this last example is actually very simple to solve by hand because so many of the coefficients are 0. In particular, the system of equations
is actually equivalent to
The first two equations tell us that
or
If we add
and therefore
We have said that the solution of a system of linear equations in matrix form
This system of equations does not have a solution; that is, there is no pair of values for x and y that can possibly satisfy both of these equations simultaneously. To see why, let's think about this system of equations graphically. Each equation represents a line in the plane. Suppose we write each of the equations in slope-intercept form. For the first equation, we solve for y and so obtain
Similarly, when solving for y , the second equation becomes
Therefore, we see that both lines have slope of
How does this play out with matrices? The corresponding coefficient matrix is
If you enter
Figure 4.4
The same ideas apply to systems of more than two equations in two unknowns, although it is more difficult to give a geometric interpretation to a system of more than three linear equations. Usually, most square matrices will have an inverse and the matrix expression
Another potential complication occurs when a system of linear equations has infinitely many different solutions, although it is also a fairly rare situation. Consider the system of equations
If you examine these two equations carefully, you will notice that the second equation is precisely twice the first. So the second equation provides no additional information about the two variables x and y . Effectively we only have one equation in the two unknowns. This is not enough information to solve for the two variables, so we cannot get a single, or unique, solution. For instance,
To see what happens algebraically, the only equation we have is
From a matrix point of view, the corresponding coefficient matrix is also singular and so it does not have an inverse. Again, fortunately, this is a very rare situation and usually arises fairly infrequently in practice. However, it does occur occasionally in the process of balancing chemical reactions, so we consider what happens and how to circumvent the difficulty in the following examples.
Example 7 When aluminum, Al, is treated with sulfuric acid
Solution Suppose we want to produce one molecule of aluminum sulfate,
Equivalently, we can think of this reaction equation as
where we have isolated our target molecule
Aluminum:
Hydrogen:
Sulfur:
Oxygen:
Notice that this is actually a system of four linear equations in the three unknowns x , y , and z .
This system is simple enough that we can approach it algebraically. The first equation immediately tells us that
or
The three values
If you look carefully at the four equations, you will notice that the fourth equation is precisely four times the third, so the last equation does not provide any new information. What this means is that only the first three equations give useful information, so that we really have only three equations in the three unknowns.
Notice that, if we tried to use matrix methods unthinkingly, we would take as the coefficient matrix
Since only a square matrix may have an inverse, there is obviously no inverse
Because we noticed that only the first three equations give useful information and the fourth equation is simply a multiple of the third equation, we can write this system of three linear equations in three unknowns in matrix form as
The solution
and so the balanced equation is
The system of equations in Example 7 was simple in the sense that it was fairly evident that the fourth equation was four times the third equation. Things are not usually quite that simple, as we illustrate in the next example.
Example 8 Sulfuric acid
Solution We will think of sodium sulfate,
which is equivalent to
where we have isolated our target molecule
Hydrogen:
Sulfur (
Oxygen:
Sodium (
As in Example 7, this is a system of four linear equations in the three unknowns. Fortunately, this system can be solved easily algebraically. The second equation gives
so that
and so
We now check that this solution
So our solution indeed satisfies all four equations and the balanced chemical reaction is
Unlike Example 7, though, none of the four equations is just a simple multiple of one of the other equations. Nevertheless, one of the four equations must be some combination of the other three in the sense that if we add the correct multiples of the first three equations, say, we would get the fourth equation. The key fact is that, with the three unknowns, we only need three pieces of information (3 equations) to solve the system (assuming that it has a solution). However, we then must check to be sure that the solution satisfies the remaining equation. The problem then becomes trying to decide which of the equations is redundant—meaning which is a combination of the other equations and so provides no new information. It is always possible to determine, algebraically, not only which equation can be written in terms of the others, but also what that relationship is. However, this is a somewhat more complicated process and we will not go into it here.
1. The accompanying table shows the life expectancy of a male or female child born in various years in the U.S.
|
|
Male |
Female |
|---|---|---|
|
1970 |
67.1 |
74.7 |
|
1975 |
68.8 |
76.6 |
|
1980 |
70.0 |
77.4 |
|
1985 |
71.1 |
78.2 |
|
1990 |
71.8 |
78.8 |
|
1995 |
72.5 |
78.9 |
|
2000 |
74.3 |
79.7 |
|
2005 |
75.2 |
80.4 |
Source: 2009 Statistical Abstracts of the U.S.
a. Find the regression line that fits each set of values as a linear function of time t in years since 1970.
b. Use the lines from part (a) to generate a system of two linear equations in two unknowns. Estimate graphically the year in which the life expectancy of males and females will be the same.
c. Solve the system of equations from part (b) algebraically, correct to two decimal place accuracy.
d. Solve the system of equations from part (b) using matrices.
e. Discuss the reasonableness of extending the two linear trends far enough for them to intersect.
2. The accompanying table shows the world record times, in seconds, for the 500 meter freestyle swimming race for men and for women in various years.
|
|
1992 |
1993 |
1994 |
1998 |
1999 |
2000 |
2001 |
2004 |
2006 |
2007 |
2008 |
|---|---|---|---|---|---|---|---|---|---|---|---|
|
Men |
|
|
21.50 |
21.31 |
|
21.21 |
21.13 |
21.10 |
20.98 |
20.93 |
20.48 |
|
Women |
24.75 |
24.23 |
|
|
24.09 |
23.59 |
|
|
|
|
|
a. Find the regression line that fits each set of values as a linear function of time t in years since 1990.
b. Use the lines from part (a) to generate a system of two linear equations in two unknowns. Estimate graphically the year in which the world record times for men and women will be the same.
c. Solve the system of equations from part (b) using matrices.
d. Discuss the reasonableness of extending the two linear trends far enough for them to intersect.
3. The accompanying table shows the world record times, in minutes: seconds, for the 1000 meter speed skating event for men and for women in various years.
|
|
1996 |
1997 |
1998 |
1999 |
2000 |
2001 |
2002 |
2005 |
2006 |
2007 |
|---|---|---|---|---|---|---|---|---|---|---|
|
Men |
|
|
|
|
|
|
|
|
|
|
|
Women |
|
|
|
|
|
|
|
|
|
|
a. Find the regression line that fits each set of values as a linear function of time t in years since 1995. (Hint: The times listed are in the minute: second format, so
b. Use the lines from part (a) to generate a system of two linear equations in two unknowns. Estimate graphically the year in which the world record times for men and women will be the same.
c. Solve the system of equations from part (b) using matrices.
d. Discuss the reasonableness of extending the two linear trends far enough for them to intersect.
4. When aluminum, , is treated with hydrochloric acid, , (which consists of one atom of hydrogen
5. When iron, , is exposed to oxygen,
6. When lye, or sodium hydroxide,
7. Pure alcohol, or ethanol,
8. Suppose that the student athletes at Brookdale College need a meal with 700 calories, 40 grams of protein, and 75 grams of carbs. As in Example 1, each ounce of meatloaf has 75 calories, 7 grams of protein, and 6 grams of carbs. Each scoop of mashed potatoes has 60 calories, 2 grams of protein, and 10 grams of carbs. Each ounce of green beans has 10 calories, 1 gram of protein, and 2 grams of carbs. How many ounces of meatloaf, how many scoops of mashed potatoes, and how many ounces of green beans are needed to fulfil this demand?
9. Suppose that the dietician at Brookdale College wants to provide a different option for students who are trying to lose weight, so a meal is to include 375 calories, 30 grams of protein, and 50 grams of carbs. As in Example 1, each ounce of meatloaf has 75 calories, 7 grams of protein, and 6 grams of carbs. Each scoop of mashed potatoes has 60 calories, 2 grams of protein, and 10 grams of carbs. Each ounce of green beans has 10 calories, 1 gram of protein, and 2 grams of carbs. How many ounces of meatloaf, how many scoops of mashed potatoes, and how many ounces of green beans are needed to fulfil this demand?
10. a. Given the set of data ,,,,, and , find the equation of the regression line using your calculator or an appropriate software package.
b. Calculate the sums needed for the two regression equations by completing the entries in the following table.
|
x |
y |
|
|
|---|---|---|---|
|
1 |
11 |
|
|
|
2 |
25 |
|
|
|
3 |
33 |
|
|
|
4 |
45 |
|
|
|
5 |
57 |
|
|
|
6 |
65 |
|
|
|
|
|
|
|
c. Use the results of part (b) to write the system of linear equations in a and b that can be used to determine the values for the unknown coefficients a and b in the regression equation.
d. Solve the system of equations from part (c). How do your results compare to what you obtained directly in part (a)?
e. In Sections 3.4 and 4.1, we suggested that you scale down large numbers, such as the full year 2000, 2001, 2002, in a set of data. Based on your calculations in part (b), can you explain why this kind of scaling is desirable? In particular, what do you think might happen to the sums if many data values, in the x 's, say, consisted of full years and if the y 's were also large numbers?
11. Repeat parts (a)−(d) of Problem 10 for the data ,,,,, and
In the last two sections, we repeatedly encountered the matrix equation
involves the product of a matrix (either A or
Figure 4.5
Figure 4.6
We now investigate what the product of a matrix and a column vector really means by looking at things graphically. First, any vector with two entries can be pictured as a line segment starting at the origin. Thus, if the vector is
In general, as shown in Figure 4.6, the vector
Figure 4.7
When we multiply a vector X and a matrix A, as in
Example 1 Given the matrix
use a calculator to find the product of the matrix A and the vector
Solution The vector X is shown in Figure 4.8; it extends from the origin to the point . Using the matrix features of a calculator, we form the product AX and so find that the new column vector
Figure 4.8
The length, or magnitude , of a vector X is the distance from the origin to the end of the vector. We write this as
to calculate the magnitude of a vector. For instance, the vector
Similarly, the magnitude of the vector
so this vector is slightly more than 5 times as long as X.
In general, the magnitude of any vector
as illustrated in Figure 4.10.
Figure 4.9
Figure 4.10
We next consider how the product of a matrix and a vector can be used to model the populations of two competing species over time. Suppose that herds of sheep and cattle are competing for the same environment, say the grass in an isolated valley in the mountains, and we wish to model the two populations over time. Initially, at time
For instance, if there were
But, the presence of the sheep acts to slow the rate of growth in the cattle population and, the more sheep there are, the slower the growth of the cattle population. To account for this, suppose that the population of cattle after one year is given by
So, if there were
after one year.
In a similar way, in the absence of cattle, suppose that the sheep population would grow by 50% per year, say, so that the number of sheep
after one year instead.
Example 2 a. Write the pair of equations giving the populations of sheep and cattle one year later based on the populations this year as a matrix equation.
b. Assume that there are 400 cattle and 700 sheep in the valley at the start of this process. Use the matrix equation from part (a) to predict the two populations after one year.
Solution a. We write the initial numbers of cattle and sheep as a vector of populations,
To set up the matrix, we must keep the order of the variables consistent in the two equations. Therefore, we rewrite the order of the terms in the second equation, so that we work with the equivalent pair of equations
The corresponding matrix of coefficients A, which is called the transition matrix , is then
The pair of population equations can then be written as the simple matrix equation
b. If the starting populations are
Notice that the number of cattle, 420, is the same value that we calculated before. We therefore see that, despite the competition, the cattle population actually increases somewhat (from 400 to 420) while the sheep population increases considerably (from 700 to 930).
Later in this section we will discuss the way that matrix multiplication is defined; for now, we only cite the results based on the use of technology.
The results of the above example raise some intriguing questions. For instance, can we figure out what happens after a second year? It is actually very simple to answer this using matrix methods. The transition matrix A connects the two populations between any two successive years:
In particular, the population vector,
Thus, the cattle population has decreased slightly (from 420 to 402), while the sheep population has grown even more dramatically (from 930 to 1269).
You might wonder what happens over the course of still more years. Does the cattle population continue to decrease and eventually die out? Does the sheep population continue to increase more and more rapidly? To answer questions such as this, we can continue to apply the same matrix analysis, but we need some better notation. We have the matrix equations:
and so on. Alternatively, we can write these matrix equations in a somewhat more suggestive manner as
and so forth. This suggests that we might be able to write these equations using powers of the matrix A with
and the matrix capabilities of a calculator, we find that
and therefore
which is the same set of population values we found previously for
We can continue to do this for each successive year, getting
Example 3 Find the populations of cattle and sheep in this valley
a. after 3 years;
b. after 4 years;
c. after 5 years.
Interpret the results of each.
Solution a. To find the populations after 3 years, we need
Consequently, the cattle population has decreased rather considerably during the third year (from 402 to 309) while the sheep population has increased by very little (from 1269 to 1283).
b. To find the populations after 4 years, we calculate
We therefore see that the cattle population has decreased very significantly while the sheep population has roughly doubled. Apparently, the growth in the sheep has come at the expense of the cattle.
c. To find the populations after 5 years, we calculate
Clearly, it is not possible for the number of cattle to be negative. Therefore, according to this mathematical model, the cattle population will have died out sometime during the fifth year and the sheep population would be more than five times the original number of 700 sheep. This may be unreasonable because there is no obvious reason why none of the original 400 cattle, let alone their subsequent offspring, will have survived for five years; after all, there is no indication of any predators that will eliminate the cattle. So perhaps the values used for the coefficients in the original model are not particularly accurate for modeling this situation or perhaps the model itself is too simple.
Another interesting issue we can study is the effect of the initial populations on the subsequent population values. Do the cattle always die out and the sheep flourish, as in Example 3, or might other situations hold? We ask you to explore this question in some of the problems at the end of this section.
Earlier in this section, we introduced a geometric interpretation for vectors with two components—they can be viewed as line segments starting at the origin. If the vector is
We also gave a geometric interpretation to the product
and so forth. How do these successive vectors compare to one another geometrically? We look at this in the next example.
Example 4 Consider the
and the initial vector
Solution We picture the initial vector
and so forth. We show these vectors in Figure 4.11. Notice that each successive vector is slightly shorter than the one before it. More importantly, notice that, although the inclination of each successive vector changes, there is a relatively large change in the angle between
Figure 4.11
The results of Example 4 are typical of what happens when you apply the successive powers of a matrix A to a starting vector X. Each successive power transforms the vector into a different vector, one usually having a different length and a different direction. But the directions usually become almost indistinguishable from one another and, quite quickly, the successive vectors soon appear to be one atop the other as they eventually all point in the same direction.
We next consider a similar example involving
Example 5 Since most households already subscribe to some form of "cable" TV service, the three providers of these services—cable, satellite, and phone—can expand only by taking customers away from one of the other services and so there is intense competition in many areas. A study in one area has found that, in any given year, of those who subscribe to cable TV service, 75% remain with cable, 10% switch to satellite, and 15% switch to phone. Also, of those who subscribe to satellite TV service, 20% switch to cable, 60% remain with satellite, and 20% switch to phone. Finally, of those who subscribe to phone TV service, 20% switch to cable, 10% switch to satellite, and 70% remain with phone. In this service area, there are currently 200,000 households who subscribe to cable service, 100,000 who subscribe to satellite service, and 150,000 who subscribe to phone service.
a. Set up the transition matrix A and the initial vector
b. Calculate the number of households in this area who will subscribe to each service next year.
c. Calculate the number of households in this area who will subscribe to each service the following year.
d. What happens in the long run in this area?
Solution a. The given information leads to the
where we express the values as decimals rather than as percentages. The initial vector for the number of subscribers to each service, in thousands, is
b. The vector
Notice that the number of cable subscribers has remained the same, but that the number of satellite subscribers has decreased and that the number of phone subscribers has increased by the same amount.
c. The vector
Notice that the number of cable subscribers again has remained the same, but that the number of satellite subscribers has decreased further and that the number of phone subscribers has increased by the same amount.
d. If we apply the transition matrix A repeatedly to the successive vectors, we find that they become closer and closer to the vector
Thus, eventually, there will be 200,000 households who subscribe to cable service, 90,000 who subscribe to satellite service, and 160,000 who subscribe to phone service.
Consider the system of two linear equations in two unknowns:
Using the approach in Section 4.1, we convert this system into the matrix equation
so that
Let's look carefully at how the product of A and X is formed to produce B. First, look at the left-hand side,
1. We take the product of 3, the first entry in the first row of matrix A, with x , the first entry in the column vector X;
2. Then, we take the product of 4, the second entry in the first row of A, with y , the second entry in the column vector X;
3. Finally, we add the two terms together to get
We perform the same three steps with the second row of matrix A consisting of the entries 5 and
Example 6 In Example 4 of Section 4.1, we converted the system of linear equations
into the corresponding matrix equation
Therefore
Explain the process by which the product of A and X is formed to produce B.
Solution Consider first the left-hand side,
1. Multiplying 5, the first entry in the first row of A, by x , the first entry in vector X;
2. Multiplying
3. Multiplying 4, the third entry of the first row of A, by z , the third entry in vector X.
4. Adding the three terms together to get
We do the same four steps with the second row of matrix A consisting of the entries 2, 4, and
Now suppose we want to multiply two square matrices A and Β of the same size to form AB in a natural way; that is, a way that is an obvious extension of what we have just done with the product of a matrix and a vector. We consider
so that
Think of the second matrix B as being composed of two different column vectors:
We multiply each of these vectors by the matrix A and combine the results into a new matrix. We diagram the process in Figure 4.12. We start (#1) by multiplying the first row of A by the first column
We then (#2) multiply the second row 5 and
This produces the first column of AB with the two entries 2 and 16.
Figure 4.12
Having multiplied matrix A by the first column
(#3):
(#4):
and so the second column of the product AB consists of the entries 57 and
This is the same result you would get using your calculator to calculate
The technique outlined above can be extended in a variety of ways. First, it can be applied to form the product of any two square matrices of the same dimensions, either 3 by 3 or 4 by 4 or larger. Second, it can be used to form the successive powers
We now discuss the meaning of the inverse
When we multiply
and it is called the multiplicative identity matrix or simply the identity matrix . To see why, consider any other
When we multiply A and I, we get
so that the product of any 2 by 2 matrix A and I is A.
Show that, for any 2 by 2 matrix A,
Furthermore, most (but certainly not all) square matrices A have an inverse matrix
For instance, the matrix
has an inverse matrix
To verify that this is indeed the inverse, we multiply this matrix by A to get
Similarly, we can check that
As we said previously, some matrices do not have inverses. For instance, the matrix
does not have an inverse. If you ask a calculator or software package to calculate
What would you expect the identity matrix for any 3 by 3 matrix to be? What property would it have? What property would the inverse
In problems
1.
2 .
3.
4.
5.
6. The mathematical model introduced in the text for cattle and sheep was based on an initial population of 400 cattle and 700 sheep. Suppose the two starting population values are interchanged so that there are initially 700 cattle and 400 sheep. Use the matrix equation to calculate the number of cattle and sheep in each of the following three years. Does the cattle population still decline and the sheep population still grow? If so, roughly how long does it now take for the cattle population to die out?
7. Repeat Problem 6 if the starting populations are 800 cattle and 500 sheep.
8. Repeat Problem 6 if the starting populations are 500 cattle and 500 sheep.
9. Consider the mathematical model for the population of cattle and sheep with the pair of equations
instead of the original pair of equations
a. What is the practical meaning of the change in the coefficient of in the first equation? What do you expect might happen with the populations over the following years because of this change?
b. Calculate the population of cattle and sheep, using the same starting population values of 400 cattle and 700 sheep, (i) after 1 year; (ii) after 2 years; (iii) after 3 years.
c. If the starting populations are 600 cattle and 300 sheep, repeat the calculations for the populations (i) after 1 year; (ii) after 2 years; (iii) after 3 years using both models.
10. In Example 5, we explored the distribution of households subscribing to different forms of cable TV service. Suppose now that there are 225,000 cable subscribers, 85,000 satellite subscribers, and 140,000 phone subscribers, though the transition matrix remains the same. Find the number of subscribers to each service
a. after one year;
b. after two years.
c. Find the limiting number of subscribers to each service as time passes.
d. Does the limiting number of subscribers seem to depend on the initial distribution or on the transition matrix?
11. Suppose that, of those who subscribe to cable TV service in some area, 70% remain with cable, 10% switch to satellite, and 20% switch to phone. Also, of those who subscribe to satellite TV service, 15% switch to cable, 60% remain with satellite, and 25% switch to phone. Finally, of those who subscribe to phone TV service, 15% switch to cable, 10% switch to satellite, and 75% remain with phone. In this service area, there are currently 200,000 households who subscribe to cable service, 100,000 who subscribe to satellite service, and 150,000 who subscribe to phone service. Find the number of subscribers to each service
a. after one year;
b. after two years.
c. Find the limiting number of subscribers to each service as time passes.
d. Does the limiting number of subscribers seem to depend on the initial distribution or on the transition matrix?
12. Repeat Problem 11 if the initial distribution of households is the same as in Problem 10.
13. The matrix
14. The matrix
15. In this problem, you will find the inverse matrix for the matrix
16. In the text, we stated that the matrix
a. Use a calculator to verify that this matrix does not have an inverse.
b. Attempt to find the inverse matrix to A using the same approach as in Problem 15. Describe what goes wrong so that the resulting system of linear equations does not have a solution.
Everything we have discussed up to this point has involved situations in which one variable (say y ) depends on another (say x ) in a linear fashion. In the real world things aren't always this simple and we often encounter situations in which one quantity depends on two or more other quantities. For example, when the weather report gives the wind-chill factor during the winter, that value depends on both the air temperature and the wind speed so it is a function of two independent variables. Similarly, when you take out a car loan, the monthly payment depends on the amount borrowed, the interest rate, and the length of the loan, so there are three independent variables. A study conducted at a college found that student performance in math classes could be accurately predicted from a combination of a dozen independent variables, including the student's score on a placement test, the student's age, gender, SAT score, high school GPA, number of years since the previous math course, and the grade in that course. A model based on all of them produced the most accurate predictions.
To keep things simple, suppose that we have a single variable y that depends on two independent variables
|
Serum Cholesterol y |
Weight |
Systolic Blood Pressure |
|---|---|---|
|
152.2 |
59.0 |
108 |
|
158.0 |
52.3 |
111 |
|
157.0 |
56.0 |
115 |
|
155.0 |
53.5 |
116 |
|
156.0 |
58.7 |
117 |
|
159.4 |
60.1 |
120 |
|
169.1 |
59.0 |
124 |
|
181.0 |
62.4 |
127 |
|
174.9 |
65.7 |
122 |
|
180.2 |
63.2 |
131 |
|
174.0 |
64.2 |
125 |
For instance, suppose that we want to determine whether a person's serum cholesterol level depends on his or her weight and systolic blood pressure and if it does, what the functional relationship is. (In a blood pressure measurement, such as 120 over 80, the systolic reading is the first, or higher, number, 120; the smaller number, 80, is the diastolic reading.) We collect a set of data on a random sample of individuals from some population group—say, young males. The data in the accompanying table, for a sample of 11 apparently normal males between the ages of 13 and 16, shows the weight of each of these individuals in kilograms (independent variable
To understand the ideas involved in finding a linear function that fits this set of data and that can then be used to answer predictive questions, we have to extend some of the ideas from the previous chapters. First, if we have any ordered pair of numbers x and y , we visualize them as representing the coordinates of a point in the plane. Similarly, suppose we have an ordered triple of numbers, say x , y , and z , where we think of z as depending on the values of x and y . Alternatively, to allow us to extend these ideas to situations with more than two independent variables, we can write the variables as
Figure 4.13
Second, recall how, in Section 3.4, we fit a line to a set of data by finding the linear function
where a , b , and c are three constants. Similarly, if the dependent variable is z and the two independent variables are x and y , the equation of a plane is
Figure 4.15.
Our objective is to find the linear function that is the best fit to the set of three-dimensional data points. That is, we want to find the plane in space that comes closest to all of the
The calculations involved in any multivariate regression are extremely tedious, but the method is so widely used that a routine is available in almost any statistical software package and in most spreadsheets and on a few hand-held calculators. We will not be concerned with the mechanics of calculating these quantities, but will simply cite the results obtained by using appropriate software. In the problems at the end of this section, we similarly assume that you have access to software that performs the calculations for you. Later in the section, we outline how to use Excel to perform such calculations.
Example 1 For the data on serum cholesterol level of young males versus their body weights
a. Find the multivariate regression equation expressing serum cholesterol level as a function of both body weight and systolic blood pressure.
b. Interpret the coefficients in the multivariate regression equation.
c. Use the regression equation to predict the serum cholesterol level of a male in the 13- to 16-year-old age group who weighs 60 kg and whose systolic blood pressure is 123.
Solution a. Using an appropriate software package, we find that the multivariate regression equation is
or equivalently
Cholesterol Level
b. To interpret this regression equation, let's see what happens to the values for the cholesterol level when we change one of the independent variables, either the weight or the systolic blood pressure. In particular, suppose that a person's weight W or
c. According to this multivariate linear regression model, we predict that the serum cholesterol level for an individual who weighs
or about 169 mg per .
Note how the coefficients
where
Also note that, in making the prediction in part (c) of Example 1, we took values for the independent variables
In addition to finding the equation of the plane that is the best fit to a set of data points in three-dimensional space, we also need a way of measuring how good the fit is. With data points (where y is a function of x ), we used the linear correlation coefficient r as such a measure. With two or more independent variables, we use a comparable quantity known as the multiple correlation coefficient , denoted by R . Like r , R takes on values between
Furthermore, the square of the multiple correlation coefficient,
As with the multivariate regression equation, both the multiple correlation coefficient and the coefficient of determination are typically calculated as part of the output of spreadsheets and statistical software.
Example 2 Find and interpret both the multiple correlation coefficient and the coefficient of determination for the data in Example 1 relating serum cholesterol level to an individual's weight and systolic blood pressure.
Solution Using an appropriate software package, we find that the coefficient of determination is
and so the multiple correlation coefficient is
The multiple correlation coefficient is reasonably close to 1, so we conclude that a high degree of linear correlation exists between the dependent variable y and the two independent variables
In Example 2 the value for the coefficient of determination
Each time we introduce an additional variable that matters, we get a value for the coefficient of determination that is closer to
We now briefly introduce the use of Excel for performing a multivariate regression analysis. In Excel the dependent variable is always denoted by Y , and the different independent variables are always denoted by X . Think of the two independent variables as
Once you have entered all the data values, click Tools on the top line and scroll down to the last entry, Data Analysis . . . (We indicate the computer displays in a different font, for emphasis.) If Data Analysis . . . doesn't appear, you will have to install the Excel Analysis ToolPak™ before proceeding. To install it, click on Tools and then select Add-ins . If Analysis ToolPak is listed, just click it to permanently install it. If Analysis ToolPak isn't listed in the Add-ins dialog box, click Browse and locate the drive and folder names and the file name Analys32.xll for the Analysis ToolPak—it usually is located in the Library\Analysis folder. When you click Data Analysis . . . , you will see a long list of available statistical procedures in alphabetical order. Scroll down until you reach Regression and then either double click it or single click and then click OK. Doing so brings up the window shown in Figure 4.17.
Figure 4.16
In this window, you first enter the Input Y range—the cells in which the values of the dependent variable y have been entered. The simplest way to do this is to click the icon at the right end of the box; this brings you back to the original spreadsheet, and you can highlight the entries down the first column under A (the y values) and then press Enter. The first box should then read: $A$1:$A$11. You then enter the Input X range—the cells in which the values of the two (or more) independent variables have been entered. Again, click the icon at the right end of the box and then highlight the entries in the second and third columns under B and C and press Enter. The second box should then show: $B$1:$C$11.
Figure 4.17
Finally, you have to enter the Output range —where the results of the regression calculations will appear on the spreadsheet. You don't want them printed over the data values in the first three columns, so you probably want them printed starting, say, in column E. Click the white circle to the left of Output range and then click the icon at the right end of the box. Designate a block of cells starting at the top of column E and extending down and to the right by highlighting the first cell under E; then press Enter. Finally, in the Regression window, press OK.
Excel will then display a large amount of information, much of which is shown in Figure 4.18. Only a few of these entries are of interest to us—the ones that have been highlighted in the figure; the rest are used for more sophisticated statistical analysis. In particular, note that the first block of output is called Regression Statistics and that the first two numbers under it are labeled Multiple R and R Square —these are the values for the multiple correlation coefficient R and the coefficient of determination,
The third block of output starts with three lines labeled Intercept, X Variable 1 , and X Variable 2 . The entries to their right give the vertical intercept and the coefficients of and , respectively. In particular, if the regression equation is
Figure 4.18
A Table vs a List The kind of table needed to perform multivariate regression is very specific in that the data values for the dependent variable and the two or more independent variables must be listed in columns or in rows, so the table is actually a list. However, we often see tabular displays for a function of two variables that are not lists and so multivariate regression cannot be applied directly. For instance, consider the following table giving values for the wind-chill factor (you will be asked to work with this in one of the Problems at the end of the Section). For each wind speed value (
|
|
Temperature |
|||||||
|---|---|---|---|---|---|---|---|---|
|
Wind speed |
35 |
30 |
25 |
20 |
15 |
10 |
5 |
0 |
|
5 |
33 |
27 |
21 |
16 |
12 |
7 |
0 |
|
|
10 |
22 |
16 |
10 |
3 |
|
|
|
|
|
15 |
16 |
9 |
2 |
|
|
|
|
|
|
20 |
12 |
4 |
|
|
|
|
|
|
|
25 |
8 |
1 |
|
|
|
|
|
|
Example 3 Rewrite the data in the accompanying table into list format so that you can apply multivariate regression.
|
|
x |
|||
|---|---|---|---|---|
|
y |
|
1 |
2 |
3 |
|
|
4 |
11 |
22 |
33 |
|
|
5 |
44 |
55 |
66 |
|
|
6 |
77 |
88 |
99 |
Solution For each value of
|
x |
1 |
1 |
1 |
2 |
2 |
2 |
3 |
3 |
3 |
|---|---|---|---|---|---|---|---|---|---|
|
y |
4 |
5 |
6 |
4 |
5 |
6 |
4 |
5 |
6 |
|
|
11 |
44 |
77 |
22 |
55 |
88 |
33 |
66 |
99 |
In this form, you can now use Excel to do the regression analysis.
1. A study was conducted relating the adult heights of women y to the heights
|
y |
|
|
|---|---|---|
|
58.6 |
63 |
64 |
|
64.7 |
67 |
65 |
|
65.3 |
64 |
67 |
|
61.0 |
60 |
72 |
|
65.4 |
65 |
72 |
|
67.4 |
67 |
72 |
|
60.9 |
59 |
67 |
|
63.1 |
60 |
71 |
|
60.0 |
58 |
66 |
|
71.1 |
72 |
75 |
|
62.2 |
63 |
69 |
|
67.2 |
67 |
70 |
|
63.4 |
62 |
69 |
|
68.4 |
69 |
72 |
|
62.2 |
63 |
66 |
|
64.7 |
64 |
76 |
|
59.6 |
63 |
69 |
|
61.0 |
64 |
68 |
|
64.0 |
60 |
66 |
|
65.4 |
65 |
68 |
Source: Mario Triola, Elementary Statistics ,
Boston: Addison-Wesley, 2010.
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient based on this sample. How much of the variation in heights of the daughters in the sample is explained by the heights of their parents?
b. Find the equation of the plane that best fits the data.
c. Use your equation from part (b) to predict the height of a daughter whose mother is 62 inches tall and whose father is 71 inches tall.
2. A study was conducted relating the heights of teenage boys y to the length x of their radius bones in the forearm and the length
|
y |
|
|
|---|---|---|
|
149.0 |
21.00 |
42.50 |
|
152.0 |
21.79 |
43.70 |
|
155.7 |
22.40 |
44.75 |
|
159.0 |
23.00 |
46.00 |
|
163.3 |
23.70 |
47.00 |
|
166.0 |
24.30 |
47.90 |
|
169.0 |
24.92 |
48.95 |
|
174.5 |
25.80 |
50.30 |
|
176.1 |
26.01 |
50.90 |
|
176.5 |
26.15 |
50.85 |
|
179.0 |
26.30 |
51.10 |
Source: Wayne W. Daniel, Biostatistics: A Foundation
for A nalysis in the Health Sciences , 4th ed., New York:
John Wiley & Sons, 1987.
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in height is explained by the length of the two bones?
b. Find the equation of the plane that best fits the data.
c. Use your equation from part (b) to predict the height of a teenage boy whose radius measures 25.50 cm and whose femur measures 49.90 cm. How close does your prediction come to the boy's actual height of 172 cm?
3. A study was conducted on the Old Faithful Geyser in Yellowstone National Park to find the relationship, if any, between the height y of the geyser (in feet) and the duration
|
Duration |
Interval |
Height |
Duration |
Interval |
Height |
|---|---|---|---|---|---|
|
240 |
86 |
140 |
218 |
78 |
140 |
|
237 |
86 |
154 |
226 |
91 |
135 |
|
122 |
62 |
140 |
250 |
89 |
141 |
|
267 |
104 |
140 |
245 |
79 |
140 |
|
113 |
62 |
160 |
120 |
57 |
139 |
|
258 |
95 |
140 |
267 |
100 |
110 |
|
232 |
79 |
150 |
103 |
62 |
140 |
|
105 |
62 |
150 |
270 |
87 |
135 |
|
276 |
94 |
160 |
241 |
70 |
140 |
|
248 |
79 |
155 |
239 |
88 |
135 |
|
243 |
86 |
125 |
233 |
82 |
140 |
|
241 |
85 |
136 |
238 |
83 |
139 |
|
214 |
86 |
140 |
102 |
56 |
100 |
|
114 |
58 |
155 |
271 |
81 |
105 |
Source: National Park Service
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in the height of an eruption is explained by the other variables?
b. Find the equation of the plane that best fits the data.
c. Which variable, the duration or the interval since the previous eruption, has a greater effect on the height of an eruption?
d. Predict the height of an eruption of Old Faithful if it lasts 200 seconds and it has been 90 minutes since the previous eruption.
e. Suppose you arrive at Old Faithful just before an eruption that lasts for 225 seconds and reaches a maximum height of 150 feet. How long has it been since the previous eruption?
4. The temperature-humidity index (THI), or simply the heat-index, combines the temperature Τ and the relative humidity Η into a single number that shows the apparent temperature I as it feels to the body. The following chart contains a selection of different combinations of the temperature and the relative humidity and the associated value for the heat index. These values are selected from a considerably more extensive table of values that can be found in any almanac or on the web.
|
Temp |
70 |
70 |
75 |
75 |
80 |
80 |
80 |
85 |
85 |
85 |
90 |
90 |
90 |
95 |
95 |
95 |
100 |
100 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Hum |
40 |
80 |
40 |
80 |
50 |
70 |
90 |
50 |
70 |
90 |
50 |
70 |
90 |
50 |
70 |
90 |
50 |
70 |
|
THI |
68 |
71 |
74 |
78 |
81 |
84 |
88 |
88 |
93 |
102 |
96 |
106 |
122 |
107 |
124 |
150 |
107 |
144 |
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in the heat-index is explained by the two independent variables?
b. Find the equation of the plane that best fits the data and so approximates the relationship between the heat-index and the temperature and the humidity. (Note: The actual formula used to calculate the heat-index is not a linear function.)
c. Use your equation from part (b) to predict the heat-index corresponding to a temperature of
d. Suppose that the air temperature is
e. According to this model, which variable, temperature or humidity, has a greater effect on the heat-index? Explain.
5. The wind-chill factor is an adjustment made to temperature readings to take into account the effects of the wind and so indicate how cold it feels. The following table gives the wind-chill factors associated with different combinations of air temperature in degrees Fahrenheit and wind speeds in miles per hour.
|
|
Temperature |
|||||||
|---|---|---|---|---|---|---|---|---|
|
Wind speed |
35 |
30 |
25 |
20 |
15 |
10 |
5 |
0 |
|
5 |
33 |
27 |
21 |
16 |
12 |
7 |
0 |
|
|
10 |
22 |
16 |
10 |
3 |
|
|
|
|
|
15 |
16 |
9 |
2 |
|
|
|
|
|
|
20 |
12 |
4 |
|
|
|
|
|
|
|
25 |
8 |
1 |
|
|
|
|
|
|
a. Select a variety of representative values (say a dozen) from this table to create a list (as we did in Example 3) having the same format as that in Problem 4 on the heat-index.
b. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in the wind-chill factor is explained by the two independent variables?
c. Find the equation of the plane that best fits the data and so approximates the relationship between the wind-chill factor and the temperature and the humidity. (Note: The actual formula used to calculate the wind-chill factor is not a linear function.)
d. Use your equation from part (c) to predict the wind-chill factor corresponding to a temperature of
e. Suppose that the air temperature is
f. According to this model, which variable, temperature or wind speed has a greater effect on the wind-chill factor? Explain.
6. The body-mass index (BMI) is a measure of the fat in the human body that is based on a person's weight and height. It is widely used by physicians to assess the risk of diseases and disabilities associated with an unhealthy weight. The following table shows the BMI for various combinations of weights (in pounds) and heights (in inches).
|
|
Weight |
||||||||
|---|---|---|---|---|---|---|---|---|---|
|
Height |
100 |
120 |
140 |
160 |
180 |
200 |
220 |
240 |
260 |
|
60 |
20 |
23 |
27 |
31 |
35 |
39 |
43 |
47 |
51 |
|
63 |
18 |
21 |
25 |
28 |
32 |
35 |
39 |
43 |
46 |
|
66 |
16 |
19 |
23 |
26 |
29 |
32 |
36 |
39 |
42 |
|
69 |
15 |
18 |
21 |
24 |
27 |
30 |
32 |
35 |
38 |
|
72 |
14 |
16 |
19 |
22 |
24 |
27 |
30 |
33 |
35 |
|
75 |
12 |
15 |
17 |
20 |
22 |
25 |
27 |
30 |
32 |
a. Select a variety of the values from this table and create a list with the same format as in Problem 4.
b. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in the BMI value is explained by the two independent variables?
c. Find the equation of the plane that best fits the data. (Note: The actual relationship between the variables is not a linear function.)
d. Use your equation from part (c) to predict the BMI corresponding to a person who weighs 175 pounds and who is 71 inches tall; for someone who weighs 205 pounds and is 71 inches tall.
e. According to this model, which variable, weight or height, has a greater effect on the BMI? Explain.
7. It has been found that the height H of waves on the open ocean depends on both the wind speed W and the duration D that the wind has been blowing at that speed. The following table shows the wave heights (in feet) for various combinations of wind speed (in miles per hour) and the duration of the wind (in hours).
|
|
Wind Speed W |
|||||
|---|---|---|---|---|---|---|
|
Duration D |
11.5 |
17.3 |
23.0 |
34.5 |
46.0 |
57.5 |
|
10 |
2 |
4 |
7 |
13 |
21 |
29 |
|
20 |
2 |
5 |
8 |
17 |
28 |
40 |
|
30 |
2 |
5 |
9 |
18 |
31 |
45 |
|
40 |
2 |
5 |
9 |
19 |
33 |
48 |
Source: Navarra, Atmosphere, Weather and Climate
a. Select a variety of the values from this table and create a table with the same format as in Problem 4.
b. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in the height of ocean waves is explained by the two independent variables?
c. Find the equation of the plane that best fits the data. (Note: The actual relationship between the variables is not a linear function.)
d. Use your equation from part (c) to predict the height of waves corresponding to a wind speed of 20 mph that has been blowing for 12 hours.
e. Suppose that the wind speed is 30 mph and the wave heights are 10 feet. Use the equation of the plane to predict how long the wind has been blowing at that speed.
f. According to this model, which variable, wind speed or duration, has a greater effect on the height of the waves? Explain.
8. The taste of cheddar cheese is related to the concentration of a number of chemicals, including acetic acid, hydrogen sulfide
|
Taste |
Acetic Acid |
|
Lactic Acid |
|---|---|---|---|
|
12.3 |
4.54 |
3.14 |
0.86 |
|
20.9 |
5.16 |
5.04 |
1.53 |
|
39.0 |
5.37 |
5.44 |
1.57 |
|
47.9 |
5.76 |
7.50 |
1.81 |
|
5.6 |
4.66 |
3.81 |
0.99 |
|
37.3 |
5.89 |
8.73 |
1.29 |
|
21.9 |
6.08 |
7.97 |
1.78 |
|
25.9 |
5.70 |
7.60 |
1.09 |
|
18.1 |
4.99 |
3.85 |
1.29 |
|
34.9 |
5.74 |
6.14 |
1.68 |
Source: Moore, David S. And George P. McCabe: Introduction to the Practice of Statistics , Freeman, 1989.
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in taste is explained by the three independent variables?
b. Find the multivariate regression equation of the hyperplane that best fits the data.
c. Use your equation from part (b) to predict the taste rating of a sample of cheddar cheese whose concentration levels of acetic acid, hydrogen sulfide, and lactic acid are 5, 6, and 1.5, respectively.
d. According to this model, which variable, acetic acid, hydrogen sulfide, or lactic acid, has the greatest effect on taste? Explain.
9. A study was conducted to find the relationship, if any, between the systolic blood pressure reading of middle-aged men and their age
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in systolic blood pressure is explained by the three variables?
b. Find the equation of the plane that best fits the data.
c. Use your equation from part (b) to predict the systolic blood pressure reading of a 54 year old man who weighs 200 pounds and who spends 8 hours a month in meditation.
|
Systolic |
|
|
|
|---|---|---|---|
|
141 |
46 |
207 |
6 |
|
153 |
47 |
215 |
1 |
|
137 |
36 |
190 |
2 |
|
139 |
46 |
210 |
1 |
|
135 |
44 |
214 |
5 |
|
139 |
45 |
226 |
9 |
|
133 |
45 |
237 |
9 |
|
150 |
56 |
229 |
7 |
|
131 |
42 |
179 |
1 |
|
146 |
53 |
217 |
10 |
10. The value of a diamond is measured by four characteristics, its carat weight, its color, its clarity, and its cut. (These are sometimes referred to as the 4C's of a diamond.) The color of a diamond is usually graded on a scale in which the letters D, E, and F indicate exceptional white coloring and subsequent letters G, H, indicate various degrees of yellowish tinting. The accompanying table shows the prices of a group of diamonds and their carat weight and color, where we have given a numerical value for each color letter, so that D is listed as 1, Ε as 2, and so forth.
|
Carats W |
Color C |
Price Ρ |
Carats W |
Color C |
Price Ρ |
|---|---|---|---|---|---|
|
0.30 |
1 |
|
0.50 |
2 |
|
|
0.35 |
2 |
|
0.60 |
6 |
|
|
0.40 |
3 |
|
0.70 |
6 |
|
|
0.50 |
2 |
|
0.75 |
1 |
|
|
0.45 |
7 |
|
1.00 |
2 |
|
|
0.60 |
3 |
|
1.00 |
5 |
|
|
0.70 |
3 |
|
1.01 |
1 |
|
|
0.80 |
7 |
|
0.20 |
1 |
|
|
0.90 |
7 |
|
|
|
|
Source: Student Project
a. Find the coefficient of determination and the multiple regression coefficient. Is there a significant level of correlation?
b. Find the equation of the regression plane that fits this data on diamond price Ρ asa function of carat weight W and color C .
c. How much of the variation in the price of a diamond can be explained by the two variables carat weight and color? What does that suggest about the relative importance of clarity and cut?
d. What is the practical significance of the fact that the coefficient of C is negative?
e. Predict the price of a diamond that weighs 0.65 carats and whose color is rated as 5 (for H).
f. Which variable, carat weight or color, has the larger impact on the price of a diamond?
11. A study was conducted to find the relationship, if any, between the amount of tar in cigarettes and the amount of nicotine they contain and the amount of carbon monoxide (CO) they have. All measurements are in milligrams per cigarette and do not include menthol or light types. The data are shown in the accompanying table.
|
Brand |
Tar |
Nicotine |
|
|---|---|---|---|
|
Benson & Hedges |
16 |
1.2 |
15 |
|
Camel |
16 |
1.0 |
17 |
|
Capri |
9 |
0.8 |
6 |
|
Carlton |
1 |
0.1 |
1 |
|
Kent |
13 |
1.0 |
13 |
|
Lucky Strike |
13 |
1.1 |
13 |
|
Marlboro |
16 |
1.2 |
15 |
|
Merit |
9 |
0.7 |
11 |
|
Newport |
11 |
0.9 |
15 |
|
Old Gold |
18 |
1.4 |
18 |
|
Pall Mall |
15 |
1.2 |
15 |
|
Raleigh |
15 |
1.0 |
16 |
|
Tareyton |
14 |
1.0 |
17 |
|
Viceroy |
18 |
1.4 |
15 |
|
Winston |
16 |
1.1 |
18 |
Source: Federal Trade Commission
a. Use an appropriate software package to calculate the coefficient of determination and the multiple correlation coefficient. How much of the variation in tar content is explained by the two variables?
b. Find the equation of the plane that best fits the data.
c. Predict the amount of tar in a cigarette that contains 1.3 mg of nicotine and 14 mg of carbon monoxide.